Plots the worldline of a "test rocket" in the accelerated reference frame of an "observer rocket" (special relativity)

RocketWorldlinesInAccelFrame.tex

\documentclass[crop=true, border=10pt]{standalone}
\usepackage{comment}
\begin{comment} 
:Title: Motion of another rocket in an accelerated rocket frame
:Slug: Accelerated frame
:Tags:  special relativity
:Author: J A Miller, UAH Physics & Astronomy, millerja@uah.edu, 2020/05/18

Worldlines of a test rocket in the accelerated coordinates $\xi$ and $\eta$ of an observer rocket. This illustrates that a constant proper distance requires different accelerations.

Figure caption:
\caption{The worldlines (\textit{purple lines}) of a second, test rocket in the accelerated reference frame of an observer rocket. The proper acceleration of the observer rocket is $\alpha=1$ and the test rocket is at an initial distance of $4$ away from the observer rocket (in both the inertial and observer rocket frames). The \textit{red dots} are the location of the of the test rocket at test rocket proper times $0$, $5$, $10$, and $15$.} 

Basic plot style and colors from:
    https://gist.github.com/mcnees/45b9f53ad371c38ba6f3759df5880fb1

---------------------------------------------------------------------
Background
---------------------------------------------------------------------

The equations of motion for an object undergoing uniform proper acceleration $\alpha$ are
\begin{equation}
    \begin{split}
		x &= \frac{1}{\alpha} \cosh(\alpha \tau) - \frac{1}{\alpha} + x_0 \\
		t &= \frac{1}{\alpha} \sinh(\alpha \tau),
    \end{split}
\end{equation}
where $\tau$ is the proper time, and $x_0$ is the starting location.

Accelerated coordinates can be defined by
\begin{equation}
    \begin{split}
		\xi &= \qty[ \qty( x - x_0 + \frac{1}{\alpha} )^2 - t^2  ]^{1/2} - \frac{1}{\alpha} \\
		\eta &= \frac{1}{\alpha} \tanh^{-1} \frac{t}{x - x_0 + 1/\alpha}.
    \end{split}
\end{equation}
When you put the equations of motion into these coordinates, you get $\xi = 0$ and $\eta = \tau$, as you would want. The object stays stationary at $\xi=0$ as the coordinate system rides with it. The quantity $\eta$ is the time coordinate, and equals the proper time. The transformation is only defined for $|t| < x - x_0 + \frac{1}{\alpha}$.

We can plot the motion of another rocket, a "test rocket", in the frame of an "observer rocket." Equivalently, these could be the two ends of the same rocket.

We see that to maintain a constant proper distance between the rockets (or the rocket ends), the acceleration must vary according to
\begin{equaiton}
	\alpha_R = \frac{\alpha_L}{1 + L_0 \alpha_L},
\end{equation}
where $L_0$ is the proper distance between the rockets and R and L denote right and left, respectively.

\end{comment}




\usepackage{tikz}
\usetikzlibrary{arrows.meta}

\usepackage{pgfplots}
\pgfplotsset{compat=1.16}

\usepackage{xcolor}
% colors
\definecolor{plum}{rgb}{0.36078, 0.20784, 0.4}
\definecolor{chameleon}{rgb}{0.30588, 0.60392, 0.023529}
\definecolor{cornflower}{rgb}{0.12549, 0.29020, 0.52941}
\definecolor{scarlet}{rgb}{0.8, 0, 0}
\definecolor{brick}{rgb}{0.64314, 0, 0}
\definecolor{sunrise}{rgb}{0.80784, 0.36078, 0}
\definecolor{lightblue}{rgb}{0.15,0.35,0.75}


% ----------------------- user specifications ------------------------

% proper acceleration of observer rocket
\newcommand*{\accel}{1.0}

% calculate the inverse of observer rocket acceleration
\pgfmathsetmacro{\accelinv}{1/\accel}

% initial starting point for the observer rocket
\newcommand*{\xinit}{0}  % this is $x_0$.

% proper acceleration of "test" rocket
\newcommand*{\acceltest}{0.2} % this is now reset below to different values

% calculate the inverse of test rocket acceleration
\pgfmathsetmacro{\accelinvtest}{1/\acceltest}

% initial starting point for the test rocket
\newcommand*{\xinittest}{4}  % this is $x_0$ for the test rocket

% define inertial frame location and time for the test rocket as a function of its proper time
\pgfkeys{/pgf/declare function={xtest(\tau) = \accelinvtest*cosh(\acceltest*\tau)-\accelinvtest+\xinittest;}}
\pgfkeys{/pgf/declare function={ttest(\tau) = \accelinvtest*sinh(\acceltest*\tau);}}

% define the accelerated coordinates of the test rocket, as a function of its proper time,
% in the frame of the observer rocket
\pgfkeys{/pgf/declare function={arctanh(\x) = 0.5*(ln(1+\x)-ln(1-\x));}}
\pgfkeys{/pgf/declare function={eta(\tau) = \accelinv*arctanh(ttest(\tau)/(xtest(\tau)-\xinit+\accelinv));}}
\pgfkeys{/pgf/declare function={xi(\tau) = sqrt((xtest(\tau)-\xinit+\accelinv)^2 - ttest(\tau)^2)-\accelinv ;}}

% Set the point style for the proper time ticks
\tikzset{
	taudot/.style={circle,draw=scarlet!70,fill=scarlet!20,inner sep=1.5pt} 
 }


% ------------------------ begin figure ------------------------------

\begin{document}

\begin{tikzpicture}[scale=1.2,domain=-2:9]
	\tikzstyle{axisarrow} = [-{Latex[inset=0pt,length=7pt]}]

	% Draw the background grid.
	\draw [cornflower!30,step=0.2,thin] (-2,-1) grid (9,5);
	\draw [cornflower!60,step=1.0,thin] (-2,-1) grid (9,5);

	% Clip everything that falls outside the grid
	\clip(-2,-1) rectangle (9,5);
	
	% Draw Axes
	\draw[thick,axisarrow] (0,-1) -- (0,5);
	\node[right,inner sep=0pt] at (0.2,4.7) {$\eta$ (Observer Rocket Worldline)};
	\draw[thick,axisarrow] (-2,0) -- (9,0);
	\node[inner sep=0pt] at (8.5,-0.3) {$\xi$};
	
	% Event horizon
	\draw[thick,dashed,scarlet] (-\accelinv,-1) -- (-\accelinv,5);
	\node[rotate=90,inner sep=0pt] at (-\accelinv-0.2,2.5) {Event Horizon of Observer};
	
	% Draw the worldlines of the test rockets
	% For x_0 = 4 of the test rocket, \acceltest = 1/5 will yield constant proper distance
	% We can use different maximum proper times depending on the test rocket acceleration
	\foreach \acceltest in {0.1}
	{
	\renewcommand{\accelinvtest}{1/\acceltest};
	\draw[domain=0:6,smooth,variable=\tau,plum,thick,samples=100,axisarrow] 
		   plot ({xi(\tau)},{eta(\tau)});
	}
	
	\foreach \acceltest in {0.3}
	{
	\renewcommand{\accelinvtest}{1/\acceltest};
	\draw[domain=0:9,smooth,variable=\tau,plum,thick,samples=100,axisarrow] 
		   plot ({xi(\tau)},{eta(\tau)});
	}
	
	\foreach \acceltest in {0.22}
	{
	\renewcommand{\accelinvtest}{1/\acceltest};
	\draw[domain=0:15,smooth,variable=\tau,plum,thick,samples=100,axisarrow] 
		   plot ({xi(\tau)},{eta(\tau)});
	}
	
	% use larger proper times for the correct one	
	\foreach \acceltest in {0.2}
	{
	\renewcommand{\accelinvtest}{1/\acceltest};
	\draw[domain=0:15,smooth,variable=\tau,plum,thick,samples=100,axisarrow] 
		   plot ({xi(\tau)},{eta(\tau)});
	}
	
	% Add the location at the text rocket proper times		   
	\foreach \tau in {0,5,10,15}		
		{
			\node[taudot] at ({xi(\tau)},{eta(\tau)});
		}

	
	% Some more labels	
	\node[right,inner sep=0pt] at (1.5,1.9) {$\alpha_t = 0.1$};
	\node[inner sep=0pt] at (4,3.2) {$\alpha_t= 0.2$};
	\node[inner sep=0pt] at (8.2,1.9) {$\alpha_t= 0.3$};
	\node[inner sep=0pt] at (7.5,2.9) {$\alpha_t= 0.22$};
	\node[right,inner sep=0pt] at (5.0,3.5) {Test Rocket Worldlines};
	

	
\end{tikzpicture}
			

\end{document}

Plot using the parameters in the above file.